Linear Algebra: Example Sheet 3 of 4

Michaelmas Term 2014
SJW
Linear Algebra: Example Sheet 3 of 4
1. Show that none of the following matrices are similar:




1 1 0
1 1 0
0 1 1,
0 1 0,
0 0 1
0 0 1
Is the matrix

1
0
0
similar to any of them? If so, which?
0
2. Find a basis with respect to which
1
−1
2
1
1
0

1
0
0
0
1
0

0
0.
1

1
1
1
is in Jordan normal form. Hence compute
0
1
−1
2
n
.
3. (a) Recall that the Jordan normal form of a 3 × 3 complex matrix can be deduced from its characteristic
and minimal polynomials. Give an example to show that this is not so for 4 × 4 complex matrices.
(b) Let A be a 5×5 complex matrix with A4 = A2 6= A. What are the possible minimal and characteristic
polynomials? If A is not diagonalisable, how many possible JNFs are there for A?
4. Let α be an endomorphism of the finite dimensional vector space V over F, with characteristic polynomial
χα (t) = tn + cn−1 tn−1 + · · · + c0 . Show that det(α) = (−1)n c0 and tr(α) = −cn−1 .
5. Let α be an endomorphism of the finite-dimensional vector space V , and assume that α is invertible.
Describe the eigenvalues and the characteristic and minimal polynomial of α−1 in terms of those of α.
6. Prove that that the inverse of a Jordan block Jm (λ) with λ 6= 0 has Jordan normal form a Jordan block
Jm (λ−1 ). For an arbitrary invertible square matrix A, describe the Jordan normal form of A−1 in terms
of that of A.
Prove that any square complex matrix is similar to its transpose.
7. Let V be a vector space of dimension n and α an endomorphism of V with αn = 0 but αn−1 6= 0. Show
that there is a vector y such that hy, α(y), α2 (y), . . . , αn−1 (y)i is a basis for V .
Show that if β is an endomorphism of V which commutes with α, then β = p(α) for some polynomial p.
[Hint: consider β(y).] What is the form of the matrix for β with respect to the above basis?
8. Let A be an n × n matrix all the entries of which are real. Show that the minimal polynomial of A over
the complex numbers has real coefficients.
9. Let V be a 4-dimensional vector space over R, and let {ξ1 , ξ2 , ξ3 , ξ4 } be the basis of V ∗ dual to the basis
{x1 , x2 , x3 , x4 } for V . Determine, in terms of the ξi , the bases dual to each of the following:
(a) {x2 , x1 , x4 , x3 } ;
(b) {x1 , 2x2 , 21 x3 , x4 } ;
(c) {x1 + x2 , x2 + x3 , x3 + x4 , x4 } ;
(d) {x1 , x2 − x1 , x3 − x2 + x1 , x4 − x3 + x2 − x1 } .
10. Let Pn be the space of real polynomials of degree at most n. For x ∈ R define εx ∈ Pn∗ by εx (p) = p(x).
Show that ε0 , . . . , εn form a basis for Pn∗ , and identify the basis of Pn to which it is dual.
11. Let α : V → V be an endomorphism of a finite dimensional complex vector space and let α∗ : V ∗ → V ∗
be its dual. Show that a complex number λ is an eigenvalue for α if and only if it is an eigenvalue for
α∗ . How are the algebraic and geometric multiplicities of λ for α and α∗ related? How are the minimal
and characteristic polynomials for α and α∗ related?
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12. (a) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear
functional θ ∈ V ∗ such that θ(x) 6= θ(y).
(b) Suppose that V is finite dimensional. Let A, B ≤ V . Prove that A ≤ B if and only if Ao ≥ B o .
Show that A = V if and only if Ao = {0}.
13. For A ∈ Matn,m (F) and B ∈ Matm,n (F), let τA (B) denote trAB. Show that, for each fixed A,
τA : Matm,n (F) → F is linear. Show moreover that the mapping A 7→ τA defines a linear isomorphism
Matn,m (F) → Matm,n (F)∗ .
14. Show that the dual of the space P of real polynomials is isomorphic to the space RN of all sequences of
real numbers, via the mapping which sends a linear form ξ : P → R to the sequence (ξ(1), ξ(t), ξ(t2 ), . . .).
In terms of this identification, describe the effect on a sequence (a0 , a1 , a2 , . . .) of the linear maps dual
to each of the following linear maps P → P :
(a) The map D defined by D(p)(t) = p0 (t).
(b) The map S defined by S(p)(t) = p(t2 ).
(c) The map E defined by E(p)(t) = p(t − 1).
(d) The composite DS.
(e) The composite SD.
Verify that (DS)∗ = S ∗ D∗ and (SD)∗ = D∗ S ∗ .
The remaining two questions are based on non-examinable material
15. Let V be a vector space of finite dimension over a field F . Let α be an endomorphism of V and let
U be an α-invariant subspace of V ie a subspace such that α(U ) ≤ U . Define α ∈ End(V /U ) by
α(v + U ) = α(v) + U . Check that α is a well-defined endomorphism of V /U .
Consider a basis hv1 , . . . , vn i of V containing a basis hv1 , . . . , vk i of U . Show that the matrix of α with
A C
respect to hv1 , . . . , vn i is
, where A the matrix of the restriction αU : U → U of α to U with
0 B
respect to hv1 , . . . , vk i, and B the matrix of α with respect to hvk+1 + U, . . . , vn + U i. Deduce that
χα = χαU χα .
16. (Another proof of the Cayley Hamilton Theorem.) Assume that the Cayley Hamilton Theorem holds
for any endomorphism on any vector space over the field F of dimension less than n. Let V be a vector
space of dimension n and let α be an endomorphism of V . If U is a proper α-invariant subspace of V ,
use the previous question and the induction hypothesis to show that χα (α) = 0. If no such subspace
exists, show that there exists a basis hv, α(v), . . . αn−1 (v)i of V . Show that α has matrix


0
−a0


 1 ...
−a1 




..
..


. 0
.
1 −an−1
with respect to this basis, for suitable ai ∈ F. Show that χα (t) = tn + an−1 tn−1 + · · · + a0 and that
χα (α)(v) = 0. Deduce that χα (α) = 0 as an element of End(V ).
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